Consider three positive integers x< y< z in arithmetic sequence.
Determine analytically all possible solutions of each of the following equations:
(I) x^{3} + z^{3} = y^{3} + 10yz
(II) x^{3} + y^{3} = z^{3}  2, with y< 116.
I didn't get the meaning of "arithmetic sequence" at first. Thanks for the additional input.
I was unable to find a solution that fit both equations. For the first equation, (1,4,7) and (3,6,9) fit. For the second equation, (5,6,7) works.
For a solution that fits both equations, I had first tried by combining the equations to get [x³=5yz1]. From this, I determined that x must end in either 4 or 9 since 5yz ends in either 0 or 5. Subtracting the two equations yields [z³y³=5yz+1]. Using a for the arithmatic sequence step leads to [3ax²+9a²x+7a³=5x²+3ax+2a²]. However, I could not find an integer solution that satisfied both equations.

Posted by hoodat
on 20070226 18:20:24 